Facility Location on High-dimensional Euclidean Spaces

Recent years have seen great progress in the approximability of fundamental clustering and facility location problems on high-dimensional Euclidean spaces, including $k$-Means and $k$-Median. While they admit strictly better approximation ratios than their general metric versions, their approximation ratios are still higher than the hardness ratios for general metrics, leaving the possibility that the ultimate optimal approximation ratios will be the same between Euclidean and general metrics. Moreover, such an improved algorithm for Euclidean spaces is not known for Uncapaciated Facility Location (UFL), another fundamental problem in the area. In this paper, we prove that for any $\gamma \geq 1.6774$ there exists $\varepsilon > 0$ such that Euclidean UFL admits a $(\gamma, 1 + 2e^{-\gamma} - \varepsilon)$-bifactor approximation algorithm, improving the result of Byrka and Aardal. Together with the $(\gamma, 1 + 2e^{-\gamma})$ NP-hardness in general metrics, it shows the first separation between general and Euclidean metrics for the aforementioned basic problems. We also present an $(\alpha_{Li} - \varepsilon)$-(unifactor) approximation algorithm for UFL for some $\varepsilon > 0$ in Euclidean spaces, where $\alpha_{Li} \approx 1.488$ is the best-known approximation ratio for UFL by Li.